EC 23 Reading and Problem Set Assignments

For class quiz next week, Sept. 15.

Please read the summary treatment in Schaum’s Outline on the Successive-Derivative Test for Optimization (4.6 on pp. 61-62).

Please also read the material on total, average, and marginal concepts in Schaum’s Outline (4.7-4.9 on pp. 62-64).

The quiz will include problem sets relating to the above.  Solved problems are illustrated in Schaum’s Outline (4.6-4.23 on pp. 67-79).

 

 

 

EC 23 Mid-term Exam

Hints:

Be sure you understand the three methods of solving a system of N linear equations: by ‘brute force’ or substitution; by finding the inverse of the matrix of coefficients; and by using determinants as per Cramer’s Rule.  Because the exam has limited time, you should go through the examples in the textbooks, and practice how to find the solution.

The exam question sheets should not be turned in with your blue books.  The questions are useful for class discussion, so please bring them with you for the class meetings after the exam.

Relax.  It’s only math.  It’s only economics.  It’s not the beginning or the end of the world.

 

Undistributed Middle Redux

How does the fallacy of the undistributed middle (FUM) arise in economics?

FUM can be exemplified as: The queen has a poodle; I have a poodle; I am queen.

This can be rewritten as: If I am queen, I have a poodle; since indeed I have a poodle, then I am queen.

FUM then boils down to hypothesizing that P implies Q, and then claiming that P is true because Q is true.

This restates what we ought to know from logic – that FUM is a formal (syllogistic) error. This error, when expressed in terms of P and Q, is that of not ruling out all the other possibilities that can produce Q.

In economics, we can think of P as the unseen but ‘true’ model, which has an implication Q. If we gather statistics to show that Q is true, can we proclaim hallelujah that P is true? No, because this would be exactly just another FUM! And yet we find that this thought process using FUM underlies many an article in economics journals (to EC 23 students: gentle reminder that this is a HW assignment).

Karl Popper would now stir from his grave, and say that humans never learn. (Does that mean that Karl was at least part alien?)

FUM is no fun.

EC 23 – Explaining the undistributed middle

The fallacy of the undistributed middle has the form:

“All A are B. All C are B. Therefore, all A are C.” For example: The queen has a poodle. I have a poodle. Therefore, I am the queen.

This example is obviously a fallacy because the conclusion does not make sense – we know from other knowledge that just because I have a poodle does not make me queen (except by accident).

But other instances of the fallacy are more subtle, and appear convincing. For example: Lazy people have no jobs. Uneducated people have no jobs. Lazy people are uneducated. In this example, the fallacy is less obvious because we also know from other knowledge that many lazy people are uneducated, and many uneducated people are lazy. It seems that our minds over-generalize to make the fallacy acceptable.

Why is the fallacy called the “undistributed middle”?

The standard form of a syllogism is as follows, by way of example (call this the Plato/mortal syllogism):

  • All men are mortal.
  • Plato is a man.
  • Plato is mortal.

The first sentence is the major premise; the second, the minor premise. The third sentence is the conclusion. (Premises and conclusion are sentences.) This particular syllogism is valid (take its validity as granted, for now; or you can prove its validity for yourself by using Venn Diagrams to map out the premises and the conclusion).

The three “terms” in the syllogism are the major, minor and middle terms.

The major term is the predicate term in the conclusion. The minor term is the subject term of the conclusion.

The middle term is the term that is neither the major nor minor term; the middle term appears twice in the premises. The middle term is also the term that “disappears” from the conclusion, because, in Tagalog, kasi nga, it is the term that mentally connects the subject and predicate in the conclusion.

In the Plato/mortal syllogism, the major term is “mortal.” The minor term is “Plato.” The middle term is “man.” The textbooks explain that in this syllogism, the middle term is “distributed” because by distribution we mean that all members of the set described by the middle term are “accounted for.” In this case, the middle term is man, and all men are referred to or accounted for in the premise or sentence that “all men are mortal.” In other words, a term is distributed if in the major premise, the adjective “all” applies to the term.

Consider now the following syllogism that is invalid because it has the fallacy of the undistributed middle:

  • All men are mortal.
  • Plato is mortal.
  • Plato is a man.

Even if we believe the conclusion that Plato is a man, it is a conclusion that does not follow from the two premises. Plato could be a dog and also mortal (and this fact would not violate the minor premise).

What happened? The minor term is still “Plato.” The major term is now “man” and the middle term is now “mortal.” Are all mortals accounted for? No, because, even if all men are mortal, the premise does not deny the possibility that other beings, such as dogs or cats, are mortal. In other words the middle term “mortal” is undistributed. “Men” is distributed, but it is now no longer the middle term.

(It would be different if we re-wrote the first sentence as “All mortals are men.” If so, the syllogism would be valid in form, even if the first sentence is technically not true. If all mortals are men, and Plato is mortal, it must be true or valid to say that Plato is a man. You can satisfy for yourself this validity by drawing the Venn Diagrams. And in this case, the middle term – mortals – is now distributed.)

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Homework assigned in class: Give an example of a fallacy of the undistributed middle in an economics argument or discussion.

EC 23 Homework – Math and Logic

Please read the following, and be prepared to discuss or take a short quiz on them.

Peter Cameron’s “Mathematics and Logic.”

John Stockstill’s “Logic.”

Lawrence Reed’s, “7 Fallacies of Economics.”

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URLs:
https://cameroncounts.wordpress.com/2010/01/03/mathematics-and-logic/
http://www.math.wichita.edu/history/topics/logic.html
https://fee.org/articles/7-fallacies-of-economics/

EC 23 Homework

Due by email June 27.  Please bring a hard copy to class. You may collaborate in groups of up to three.

1.      Spell out in words an economic problem, the solution for which involves math. Such a solution may involve geometry, graphs, algebra, calculus, or even a use of a property of numbers.
2.      If you have a solution, keep it to yourself. Be prepared to explain whether you understand the solution. It’s ok not to have a solution, but you have to be prepared to challenge your classmates to give their solutions.

Hints: The problem may be a small almost trivial one, but should be interesting enough for class discussion. You may use any materials you can find in books or the Internet. You may seek help from students who have already taken the course on mathematical economics.

Just for illustration, here’s an example of such a problem from consumer theory:

“Define convexity in relation to indifference curves as representations of the utility function of a consumer. What might be the properties of these indifference curves if the goods considered are perfect substitutes? What if the goods are complements? What if the goods are imperfect substitutes?”

Extra credit for valor. You may submit the illustrative problem given above, but only if you have answers that you can defend in class.

Extra credit also for submissions of an easy but very illuminating question that shows the use of math in economics.